# Volatility Density Forecasting & Model Averaging Imperial College Business School, London

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1 Imperial College Business School, London June 25, 2010

2 Introduction Contents Types of Volatility Predictive Density Risk Neutral Density Forecasting Empirical Volatility Density Forecasting Density model averaging Bayesian Approximation using Ranking Measures Thick Model Averaging

3 Data Contents Results Risk Neutral Volatility Density Forecasting Risk Neutral to Objective Volatility Density Forecasting Real World Volatility Density Forecasting Investment Strategy Results - Investment Strategy Conclusions

4 Introduction Forecasting volatility of a financial instrument has been the center of scientific research for almost three decades. If we allow more attributes to play a roll in the forecasting density then predicting directly that density and not characteristics of it becomes the imminent task to perform. Why density forecasting? Complicated structure derivative products need to deviate from log normal distributional assumption of the underlying. Cutting edge portfolio optimization strategies. Strategic Risk Management. point or interval estimation are inadequate tools for those responsible for decision making.

5 Types of Volatility Predictive Density Two distinct schools of thought addressing the problem of volatility density forecasting Real World Density Forecasting. Reflects the dynamics of real prices. Risk Neutral Density Forecasting. Predictive densities are obtained from a set of option prices.

6 Types of Volatility Predictive Density cont. Implied Volatility Density Forecasting. Breeden & Litzenberger (1978) established a relationship between the second derivative of the option price with respect to the strike price and the risk neutral density. Empirical Volatility Density Forecasting. GARCH-type Bootstrap related methods

7 How devastating a wrong model selection could be to the final result? Model misspecification risk jeopardizes the forecasting performance. Solution to the problem is to use Model Averaging Techniques. Two methods: Thick Model Averaging (Granger and Jeon, 2004). Bayesian Approximation.

8 Risk Neutral Density Forecasting Ross (1976) and Cox and Ross (1976) the call price equals: c (S T ) = e rt K S T f Q t (S T ) ds T e rt Kp Q Breeden and Litzenberger: ft Q (S T ) = e rt 2 c t (S T, K, T, t) K 2 K=ST

9 Risk Neutral Density Forecasting cont. To approximate the above density function, ft Q to formulate a butterfly spread: (S T ), we have e rt 2c (S t, K, T, t) + c (S t, K s, T, t) + c (S t, K + s, T, t) s 2 P (S t, K, T, t) lim s 0 s 2 e rt ft Q (S T )

10 Risk Neutral Density Forecasting cont. From the above is apparent the need to have a reliable OPTION PRICING MODEL. The literature in inundated with various suggestions. Of the most important: The analytic pricing formula of Grunbichler and Longstaff a Square Root Mean Reverting Process. dv T = (α κv T ) dt + σ V T dz T

11 Risk Neutral Density Forecasting cont. Based on Feller (1951) and Cox and Ross (1985) they proved that the risk adjusted underlying asset is noncentrally χ 2 (ν, λ) distributed. This gives the analytical solution to the PDE of the option pricing equation which is the following: c (V T, K) = D T E Q [ (V T K) +] [ c (V t, K) = e rt e βt V t Q (γk ν + 4, λ) + α ( β 1 e βt ) Q (γk ν + 2, λ) e rt KQ (γk ν, λ) ]

12 Risk Neutral Density Forecasting cont. The Log Mean Reverting Gaussian Process model of Detemple and Osakwe: d ln V T = (α λ ln (V T )) dt + σdz T V T = e rt V φt + η2 λ 2 N (δ + η) KN (δ) 0 e α(1 φ t )

13 Risk Neutral Density Forecasting cont. The risk adjusted Jump Diffusion Mean Reverting model of Clewlow and Strickland log V t = ( α (µ r log V t ) 1 2 σ2) t + σ tε 1t + (µ jump + σ jump ε 2t κµ jump t) (u t > κ t) ε 1i, ε 2i N { (0, 1) 1 ui > κ t u i U (0, 1) where u i 0 otherwise

14 Risk Neutral Density Forecasting cont. In a spirit similar to Schwartz (1997) (non constant volatility component for the equation that describes the underlying) as well as Detemple and Osakwe (2000) (log Ornstein Uhlenbeck model) we employ a Stochastic Volatility Double Mean Reverting process (both the underlying and the volatility process mean revert). d log V t = α (µ r log V t ) dt + σ t dw 1t d log σ 2 t = η ( ϑ log σ 2 t ) dt + γdw2t Barndorff Nielsen and Shephard model (2001) alternatives. Capture the driving dynamics of volatility options using a Gamma Ornstein Uhlenbeck process with stochastic time change.

15 Risk Neutral Density Forecasting cont. dy t = λy t dt + dz λt The process here represents the volatility of the underlying and can be simulated as: y tn = (1 λ t) y tn 1 + N tn κ=n tn 1 +1 x κ e λ tuκ pricing equation equals, log V t = ( α (µ r log V t ) 1 2 σ2 t ) t + σ t tε1t

16 Risk Neutral Density Forecasting cont. Central Tendency (time dependent mean) with Jumps µ t = c + bµ t 1 + δε t c = θ ( 1 e α t) b = e α t The underlying follows OU jump process with time dependent mean: log V t = ( α (µ t r log V t ) 1 2 σ2) t + σ tε 1t + (µ jump + σ jump ε 2t κµ jump t) (u t > κ t)

17 Transforming Risk Neutral to Real World Density Ait-Sahalia & Lo (2000) The stochastic discount factor for all options is a random variable that equals: ζ (S T ) = e r(t t) f Q (S T ) f P (S T ) = λu (S T ) U (S t ) The relative risk aversion equals, RRA = x ζ t (x) ζ t (x) U (x) = x U (x)

18 Transforming Risk Neutral to Real World Density cont. Power utility assumption here: { x 1 γ U (x) = 1 γ γ 1 log (x) γ = 1 Then the Relative Risk Aversion equals: RRA = x U (x) γx γ 1 = x U (x) x γ = γ f P (x) = 0 x γ f Q (x) f Q (y) y γ dy

19 Transforming Risk Neutral to Real World Density cont. In steps the procedure is described as follows: 1. Select an option pricing model. To avoid problems related with model uncertainty later we will use model averaging procedures instead of making such a decision. 2. Calibrate the parameters of the model using observed option prices. 3. Using a butterfly spread to approximate the 2nd derivative of the price with respect to strike obtain the empirical risk neutral predictive distribution of the underlying (the VIX index) for a month ahead.

20 Transforming Risk Neutral to Real World Density cont. 4. Obtain random draws from the empirical predictive density in order to calibrate the Gamma and the log-normal distribution that we want to use in addition to the empirical. 5. Transform the risk neutral distribution (either the empirical, Gamma or log-normal) to the objective one according to Bliss and Panigirtzoglou method i.e. connect the investors risk preferences (using the power utility function for example) with the risk neutral real world density. 6. The risk premium is obtained by maximizing the Berkowitz s test p-value related with the realizations of the index with respect to the transformed predictive distribution. 7. Repeat steps 1 to 4 for the whole sequence of forecasts.

21 Empirical Volatility Density Forecasting To account for some of the most widely accepted characteristics of conditional volatility the model that have been used in the analysis are: Model Variance Equation GARCH p q h t = ω + a i ε 2 t i + β j h t j Absolute Value GARCH ht = ω + GJR h t = ω + i=1 j=1 p a i ε t i c + i=1 q β j ht j j=1 p α i ε 2 t i + γε2 t i I {ε t i < 0} + i=1 q β j h t j j=1

22 Empirical Volatility Density Forecasting cont. TGARCH h t = ω + p a i ε t i c + i=1 q β j ht j j=1 Asymetric Power GARCH ( δ ) 2 p ht j = ω + a i ε t i δ γ ε t i δ I { ε t i < 0 } i=1 + q ( ) δ β j ht j j=1

23 Empirical Volatility Density Forecasting cont. The error parameters are allowed to be distributed less restrictively: Normal Distribution Generalized Normal Distribution Student s-t distribution Skew-t distribution To simulate a sufficient large number of variance forecasts for time t we take n independent random draws from the error distribution and add them to the volatility equation.

24 Empirical Volatility Density Forecasting cont. For each draw z t p,i i = 1, 2..., n we estimate ε t p,i i = 1, 2..., n. Next forecast h t,i i = 1, 2..., n. At the end we have a volatility density estimate for time t for every univariate conditional volatility model with lags (p, q) and marginal distribution the same one as that of the error distribution.

25 Empirical Volatility Density Forecasting cont. In steps the procedure is described as follows: 1. Select a model to forecast volatility and select an error distribution to use for that model. 2. Using return data until t p (p being the lag value) we calibrate volatility parameters using maximum likelihood. Notice that depending on the error distribution the log likelihood considered necessary to calibrate that volatility model should adjust accordingly. 3. Take n random draws from that selected error density to simulate values for z t p.

26 Empirical Volatility Density Forecasting cont. 4. Estimate ε t p,i = h t p,i z t p,i i = 1, 2..., n 5. Create n independent volatility forecasts for each time step using the simulated error values. 6. Fit the volatility sample to a choice of three distributions. First to the empirical distribution using a kernel regression, second to the Gamma and third to the log-normal distribution. 7. Asses the accuracy of each model using the Berkowitz test to the whole sequence of forecasts. The model that generates the highest p-value of the test is the optimal model to forecast the volatility density of the index.

27 Density model averaging Bayesian Model Averaging Bayesian Inference Let y = {y 1, y 2,... y n } iid observations with θ unknown parameters and prior distribution π (θ η). Inferences for θ can be made based on the posterior distribution p(θ y, η) = p(y,θ η) p(y η) = p(y,θ η) p(y,θ η)dϑ = p(y θ)p(θ η) p(y θ)p(θ η)dϑ = l(y θ)p(θ) m(y η) MCMC algorithm used to simulate posterior parameters distributions through sampling from a Marcov Chain that has as its unique stationary distribution the probability distribution of interest.

28 Bayesian Approximation using Ranking Measures The estimation of the marginal distribution difficult to perform. Kass & Wasserman (1995) used the Schwartz Information Criterion to approximate the Bayes Factor: log BF ij L Mi L Mj + k M i k Mj 2 log p

29 Bayesian Approximation using Ranking Measures cont. To perform averaging: Compute for each model available its Ranking Measure (BIC, AIC, MSE etc). The approximated marginal distribution equals: m = RM Mi Calculate posterior probabilities by dividing the approximated marginal distribution of each model with the sum of marginals of all models.

30 Thick Model Averaging Simple Thick Model Average (Granger and Jeon, 2003): Rank models according to an error criterion (eg BIC, AIC, Pesaran, Zaffaroni, 2006). Select a percentage of them (most commonly used is 10% to 50% of best performing models) and average them.

31 Risk Neutral Volatility Density Averaging Bayesian Approximation A collection of alternative competing pricing proposal is used. Each period these models are fitted to the real option data and the Mean Squared Error of that fit is reported as the first RM. ( ) N C n,t 2 N C n,t en,t 2 n=1 n=1 MSE t = = N N

32 Bayesian Approximation cont. Additional Ranking Measures are the Akaike and Bayesian Information Criteria. AIC t = e 2k N t BIC t = N k N 1 N N n=1 1 N n=1 en,t 2 = e 2k N t MSE t N en,t 2 = N k N MSE t

33 Bayesian Approximation cont. All models will be weighted according to their Bayes Factors. BF Mi,t BF Mi,t BF Mi,t BIC M i,t n BIC Mi,t i=1 AIC M i,t n AIC Mi,t i=1 MSE M i,t n MSE Mi,t i=1 i = 1,..., n i = 1,..., n i = 1,..., n

34 Thick Model Averaging Uses only a percentage of the best competing models. w Mi,t = w Mi,t = w Mi,t = p BIC Mi,t i=1 p p AIC Mi,t i=1 p n MSE Mi,t i=1 p i = 1,..., p i = 1,..., p i = 1,..., p p is the number of models included in the averaging scheme depending on the keeping percentage here p = 33%n or p = 50%n

35 Empirical Volatility Density Averaging Bayesian Approximation Estimate GARCH-type models using the maximum likelihood. Calculate relevant Ranking Criteria. ( ) N h Mi,t 2 h Mi,t i=1 MSE Mt = = N N em 2 i,t i=1 N i = 1,..., n AIC Mt = 2L Mi,t + 2 N k M i,t i = 1,..., n BIC Mt = 2L Mi,t + log(n) N k M i,t i = 1,..., n

36 Data - Risk Neutral Volatility Density Forecasting The VIX index from 6/2010 until 10/ surfaces - monthly frequency. Every month a new option surface was acquired and based on that a forecast for the next month s implied volatility density was made. 32 surfaces were discharged from the sample to calibrate the Risk Premium (following Bliss and Panigirtzoglou). A rolling window was used to recalibrate the risk premium every month. The last 12 remaining surfaces were used for density forecasting. Using the option prices of every month, forecasts for the implied volatility density of next month were made.

37 Data - Risk Neutral Volatility Density Forecasting cont. Filtering was applied to the option prices so that there were disregarded: Options with small maturity (less than ten days). Options with more than 365 days maturity. Options with value smaller than \$0.05. Options that had zero volume.

38 Data - Empirical Volatility Density Forecasting The S&P 500 index from 3/1950 to 10/ data points - monthly frequency. 12 realizations constitute the out of sample period. Actual volatility: high-low estimator (Bollen and Inder, 2002) ĥ = (ln P H t ln P Lt ) 2 4 ln 2

39 Results - Risk Neutral Volatility Density Forecasting Berkowitz test was used (Berkowitz, 2001). The additional advantage of this indicator is that both independence and uniformity are jointly tested with this test. The optimal model is the one that produced the highest test p-value.

40 Results - Risk Neutral Volatility Density Forecasting cont. Rank Model P-value Berkowitz Rank Model P-value Berkowitz 1 TM-MSE 33% Log-Normal TM-MSE 50% Gamma TM-AIC 33% Log-Normal TM-BIC 33% Emprirical TM-BIC 50% Log-Normal BA-MSE Gamma TM-MSE 50% Log-Normal TM-BIC 33% Gamma TM-AIC 50% Log-Normal BA-AIC Gamma TM-MSE 50% Emprirical BA-BIC Gamma TM-BIC 50% Emprirical BA-MSE Emprirical TM-BIC 33% Log-Normal BA-AIC Emprirical TM-MSE 33% Emprirical BA-BIC Emprirical TM-AIC 33% Emprirical BEST MSE Log-Normal TM-AIC 50% Emprirical BEST MSE Emprirical BA-MSE Log-Normal BEST BIC Log-Normal BA-AIC Log-Normal BEST AIC Log-Normal BA-BIC Log-Normal BEST AIC Emprirical TM-MSE 33% Gamma BEST BIC Emprirical TM-AIC 33% Gamma BEST MSE Gamma TM-BIC 50% Gamma BEST BIC Gamma TM-AIC 50% Gamma BEST AIC Gamma

41 Results - Risk Neutral Volatility Density Forecasting cont. Catholic dominance of the averaging schemes. Top performer is Thick Model Averaging. Assumption of either a log-normal or an empirical distributional pattern for the volatility process is a more preferable option (as opposed to a gamma equivalent). The highest p-values are recorded for option pricing methods that assume a process for the underling that exhibits either jumps (with constant mean and variance) or has a stochastic volatility dynamic (without jumps).

42 Results - Risk Neutral Volatility Density Forecasting cont. For Best Single Model strategy the log-normal or the empirical density are still more preferable choices than the gamma distribution for the volatility. In total Best Single Proposals fail to solidify their status.

43 Results - Risk Neutral to Objective Volatility Density Forecasting Rank Model P-value Berkowitz Rank Model P-value Berkowitz 1 TM-AIC 33% Emprirical BEST BIC Gamma TM-MSE 50% Emprirical BEST BIC Log-Normal TM-AIC 33% Log-Normal BEST BIC Emprirical TM-MSE 50% Log-Normal TM-AIC 50% Gamma TM-BIC 33% Emprirical TM-BIC 50% Gamma TM-MSE 33% Emprirical BEST MSE Gamma TM-AIC 50% Emprirical BEST MSE Log-Normal TM-MSE 33% Log-Normal BEST MSE Emprirical TM-AIC 50% Log-Normal BA-BIC Emprirical TM-BIC 50% Log-Normal BA-AIC Emprirical TM-BIC 50% Emprirical BA-MSE Emprirical TM-MSE 50% Gamma TM-MSE 33% Gamma TM-BIC 33% Log-Normal BA-BIC Log-Normal TM-AIC 33% Gamma BA-AIC Log-Normal BEST AIC Gamma BA-MSE Log-Normal BEST AIC Log-Normal BA-BIC Gamma BEST AIC Emprirical BA-AIC Gamma TM-BIC 33% Gamma BA-MSE Gamma

44 Results - Risk Neutral to Objective Volatility Density Forecasting cont. Averaging schemes still prevail to the single modeling alternatives. Thick Modeling under all different ranking criteria and keeping percentages of the best in sample models dominates again. For Best Single Model strategy the Gamma density is now more preferable (inconsistent performance).

45 Results - Real World Volatility Density Forecasting Empirical Distribution Log-Normal Gamma Rank Volatility Density Model P-value Volatility Density Model P-value Volatility Density Model P-value 1 TM 40% AIC TM 40% AIC TM 40% AIC TM 40% BIC TM 40% BIC TM 40% BIC TM 30% AIC TM 50% AIC TM 50% AIC TM 30% BIC TM 50% BIC TM 50% BIC TM 50% AIC TM 30% AIC TM 30% AIC TM 50% BIC TM 30% BIC TM 30% BIC TM 20% AIC TM 20% BIC TM 20% AIC TM 20% BIC TM 20% AIC TM 20% BIC BA AIC BA AIC TM 10% AIC BA BIC BA BIC TM 10% BIC TM 10% AIC TM 10% AIC BA AIC TM 10% BIC TM 10% BIC BA BIC BEST AIC BEST AIC BEST AIC BEST BIC BEST BIC BEST BIC 0.000

46 Results - Real World Volatility Density Forecasting The overwhelming majority of the methods fail to pass the Berkowitz test. Only Thick Model Averaging produces accepts the hypothesis that the predictive density is the true generating process. Log-Normal volatility density produces higher acceptance rates. Best Single Models are still the worst performers.

47 Investment Strategy Relates only with the option based predictive volatility density. VIX is considered the fear index i.e. an indicator of the movement (not the direction) of the S&P500 index. The forecasted distribution of VIX is used to bet on the equity index.

48 Investment Strategy cont. Take the current price of VIX. Derive the cumulative probability using the different predictive densities we obtained from the various option models. If the probability surpass a certain threshold we take a short position on the S&P500 index. If it fails to exceed that threshold we take a long position on the index. The threshold is obtained dynamically by optimizing the cumulative wealth of the strategy described above over the last 5 months. at each step the threshold is recalibrated.

49 Results - Investment Strategy Risk Neutral Volatility Density Forecasting Rank Model cum ret Rank Model cum ret 1 BEST AIC Emprirical TM-BIC 50% Gamma BA-AIC Emprirical BEST MSE Gamma TM-AIC 33% Emprirical BA-MSE Gamma TM-AIC 50% Emprirical TM-MSE 33% Gamma BEST BIC Emprirical TM-MSE 50% Gamma BA-BIC Emprirical BEST AIC Log-Normal TM-BIC 33% Emprirical BA-AIC Log-Normal TM-BIC 50% Emprirical TM-AIC 33% Log-Normal BEST MSE Emprirical TM-AIC 50% Log-Normal BA-MSE Emprirical BEST BIC Log-Normal TM-MSE 33% Emprirical BA-BIC Log-Normal TM-MSE 50% Emprirical TM-BIC 33% Log-Normal BEST AIC Gamma TM-BIC 50% Log-Normal BA-AIC Gamma BEST MSE Log-Normal TM-AIC 33% Gamma BA-MSE Log-Normal TM-AIC 50% Gamma TM-MSE 33% Log-Normal BEST BIC Gamma TM-MSE 50% Log-Normal BA-BIC Gamma S&P TM-BIC 33% Gamma 1.284

50 Results - Investment Strategy Risk Neutral to Objective Volatility Density Forecasting Rank Model cum ret Rank Model cum ret 1 BA-AIC Emprirical TM-AIC 33% Gamma BA-BIC Emprirical TM-AIC 50% Gamma BA-MSE Emprirical BEST BIC Gamma BA-AIC Gamma TM-BIC 33% Gamma BA-BIC Gamma TM-BIC 50% Gamma BA-MSE Gamma BEST MSE Gamma BA-AIC Log-Normal TM-MSE 33% Gamma BA-BIC Log-Normal TM-MSE 50% Gamma BA-MSE Log-Normal BEST AIC Log-Normal BEST AIC Emprirical TM-AIC 33% Log-Normal TM-AIC 33% Emprirical TM-AIC 50% Log-Normal TM-AIC 50% Emprirical BEST BIC Log-Normal BEST BIC Emprirical TM-BIC 33% Log-Normal TM-BIC 33% Emprirical TM-BIC 50% Log-Normal TM-BIC 50% Emprirical BEST MSE Log-Normal BEST MSE Emprirical TM-MSE 33% Log-Normal TM-MSE 33% Emprirical TM-MSE 50% Log-Normal TM-MSE 50% Emprirical S&P BEST AIC Gamma 1.284

51 Results - Investment Strategy Summary Results indicated a higher cumulative wealth under the Bayesian Approximation scheme against all other alternatives. Thick Modeling exhibited equal performance when compared with some single models while the method outperformed other modeling suggestions. All averaging schemes are preferable to investing directly on the S&P500 index. The worst cumulative wealth was generated by single models.

52 Conclusions Uncertainty of model selection since it is based on the in sample performance according to an error criterion. Out of sample, environment much more volatile thus initial model choice could be rendered suboptimal. Forecasting performance of the best single model out of sample, selected accurately or not in advance, enhanced when attempting averaging. Thick modeling was the best averaging alternative. All in all averaging alternatives outperformed single models. Empirical results as well (investment strategy) substantiated the superiority of averaging schemes. Method has potential for applications not only to volatility density but to any density forecasting aspiration.

53 Acknowledgements Dr. Paolo Zaffaroni Prof. Nigel Meade ISF Committee

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